The NBA rules don’t allow it. In its rule book, the N.B.A. requires players, coaches and trainers “to stand and line up in a dignified posture along the sidelines or on the foul line” during the national anthem.
The NFL has no such rule. However, there are two rules that could be stretched to apply to this situation:
Rule 5.8: “Throughout the period on game-day that a player is visible to the stadium and television audience (including in pregame warm-ups, in the bench area, and during postgame interviews in the locker room or on the field), players are prohibited from wearing, displaying, or otherwise conveying personal messages either in writing or illustration, unless such message has been approved in advance by the League office. Items to celebrate anniversaries or memorable events, or to honor or commemorate individuals, such as helmet decals, and arm bands and jersey patches on players’ uniforms, are prohibited unless approved in advance by the League office. All such items approved by the League office, if any, must relate to team or League events or personages. The League will not grant permission for any club or player to wear, display, or otherwise convey messages, through helmet decals, arm bands, jersey patches, or other items affixed to game uniforms or equipment, which relate to political activities or causes, other non-football events, causes or campaigns, or charitable causes or campaigns. Further, any such approved items must be modest in size, tasteful, non-commercial, and non-controversial; must not be worn for more than one football season; and if approved for use by a specific team, must not be worn by players on other teams in the League.”
An argument can be made that sitting or kneeling during the National Anthem is a behavioral “illustration” that is intended to convey a personal/political message in violation of the rule.
Rule 5.9: “Consistent with the equipment and uniform rules, players must otherwise present a professional and appropriate appearance while before the public on game-day. Among the types of activity that are prohibited are use of tobacco products (smokeless included) while in the bench area and use of facial makeup.”
An argument can also be made that sitting or kneeling during the National Anthem creates an unprofessional and inappropriate appearance.